Centrifugal gyroscopic devices, and associated systems and methods

ABSTRACT

Centrifugal gyroscopic devices are described herein. A representative device can include a shaft, an arm coupled to the shaft, a rotor coupled to the arm, and a control system operably coupled to the shaft, the arm, and/or the rotor. The shaft is rotatable about a first axis and the arm is configured to rotate with the shaft. The arm is pivotable about a second axis and the rotor is configured to pivot with the arm about the second axis. The rotor is further pivotable about a third axis. The control system is configured to bring the shaft, the arm, and the rotor into a resonant mode in which the shaft rotates at a rotational rate, the arm oscillates about the second axis at a first frequency substantially equal to the rotational rate, and the rotor oscillates about the third axis at a second frequency substantially equal to the first frequency.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims the benefit of U.S. Provisional PatentApplication No. 63/332,196, filed on Apr. 18, 2022, and titled“CENTRIFUGAL GYROSCOPIC DEVICES, AND ASSOCIATED SYSTEMS AND METHODS,”which is incorporated herein by reference in its entirety.

TECHNICAL FIELD

The present technology is directed to centrifugal gyroscopic devices forgenerating power and/or efficiently summing energy, and associatedsystems and methods.

BACKGROUND

U.S. Pat. No. 5,457,993, titled “Pendulous Oscillating GyroscopicAccelerometer,” describes a pendulous oscillating gyroscopicaccelerometer. The accelerometer utilizes the principle that agyroscopic torque is generated when an oscillating gyroscope isoscillated about a transverse axis. When the gyroscopic torque isbalanced by a pendulous torque, a measurement of acceleration isprovided. If the accelerometer is attached to the earth, gravity ismeasured. As such, the accelerometer operates as a sensing device.

BRIEF DESCRIPTION OF THE DRAWINGS

Many aspects of the present technology can be better understood withreference to the following drawings. The components in the drawings arenot necessarily to scale. Instead, emphasis is placed on clearlyillustrating the principles of the present technology.

FIGS. 1A and 1B are partially-schematic isometric views of a centrifugalgyroscopic device continued in accordance with embodiments of thepresent technology.

FIG. 1C is a side-cross sectional view of the centrifugal gyroscopicdevice taken in a plane extending along a first axis and a second axisof the centrifugal gyroscopic device in accordance with embodiments ofthe present technology.

FIG. 2 is an isometric view of a representative arm assembly and armmotor assemblies of the centrifugal gyroscopic device in accordance withembodiments of the present technology.

FIG. 3 is an isometric side view of a representative arm assembly of thecentrifugal gyroscopic device with a housing of the arm assembly removedin accordance with embodiments of the present technology.

FIG. 4 is block diagram illustrating physical properties/forces actingon a centrifugal gyroscopic device during operation in accordance withembodiments of the present technology.

FIGS. 5A and 5B are graphs illustrating components of an oscillatoryrotor angular momentum and an oscillatory arm assembly angular velocityof the centrifugal gyroscopic device over time in accordance withembodiments of the present technology.

FIG. 6 is a flow diagram of a method or process for operating acentrifugal gyroscopic device to generate power in accordance withembodiments of the present technology.

FIG. 7 is a perspective side view of a centrifugal gyroscopic devicecontinued in accordance with additional embodiments of the presenttechnology.

FIGS. 8A-8D are enlarged perspective views of the centrifugal gyroscopicdevice of FIG. 7 , illustrating the movement of an arm assembly androtors of the centrifugal gyroscopic device during a complete revolutionof the arm assembly in a resonant mode of operation, in accordance withembodiments of the present technology.

FIG. 9 is a schematic diagram of a control assembly for controlling aspindle in accordance with embodiments of the present technology.

FIG. 10 is a schematic diagram of a control assembly for controlling aspindle in accordance with additional embodiments of the presenttechnology.

FIG. 11 is a schematic diagram of an analog control assembly forcontrolling a spindle in accordance with additional embodiments of thepresent technology.

FIG. 12 is a graph illustrating normalized net power versus the phaseangle between rotor and arm oscillations.

FIG. 13A is a graph illustrating sample test results for the centrifugalgyroscopic device of FIGS. 1A-1C showing the change in mechanical powerof the spindle versus the amplitude of oscillation of the rotors.

FIG. 13B is a graph illustrating sample test results for the centrifugalgyroscopic device of FIGS. 1A-1C showing the extraction of gyroscopicpower from spindle rate harmonics.

FIGS. 14A-14C are graphs illustrating sample test results for thecentrifugal gyroscopic device of FIGS. 1A-1C.

FIGS. 15A-15D are graphs illustrating sample test results for theeffects of oscillation of the rotors on movement of the arm assembly ofthe centrifugal gyroscopic device FIGS. 1A-1C.

FIGS. 16A-16F are graphs illustrating sample test results for theeffects of oscillation of the arm assembly on the torque of the rotorsof the centrifugal gyroscopic device of FIGS. 1A-1C.

DETAILED DESCRIPTION

Aspects of the present disclosure are directed generally towardcentrifugal gyroscopic devices for generating power and/or efficientlysumming energy, and associated systems and methods. In several of theembodiments described below, a representative centrifugal gyroscopicdevice includes (i) a shaft, (ii) an arm coupled to the shaft, (iii) atleast one rotor coupled to the arm, and (iv) a control system operablycoupled to at least one of the shaft, the arm, and the rotor. The shaftis rotatable about a first axis and the arm is configured to rotate withthe shaft. The arm is pivotable about a second axis different from thefirst axis, and the at least one rotor is configured to pivot with thearm about the second axis. The at least one rotor is further pivotableabout a third axis different from the first and second axes. The controlsystem is configured to bring the shaft, the arm, and the rotor into aresonant mode of operation in which (a) the shaft rotates at arotational rate, (b) the arm oscillates about the second axis at a firstfrequency generally equal to the rotational rate, and (c) the at leastone rotor oscillates about the third axis at a second frequencygenerally equal to the first frequency. Energy can be input into thedevice via the control system to control the motion of the shaft, thearm, and the rotor; and energy can be output from the device via theshaft, such as via a power generator coupled to the shaft and configuredto convert the mechanical energy from the rotation of the shaft intoelectrical energy.

During operation, the rotation of the shaft rotates the arm andgenerates a centrifugal force that acts against the arm to oscillate thearm about the second axis. The oscillating motion of the arm and the atleast one rotor combine to generate a gyroscopic torque that acts torotate the shaft about the first axis. In some embodiments, the controlsystem is configured to change a phase relationship between thefrequency of the oscillation of the arm about the second axis and thefrequency of the oscillation of the at least one rotor about the thirdaxis to change an average value of the gyroscopic torque. For example,the control system can drive the arm and/or the at least one rotor viaone or more motor assemblies to bring the frequency of the oscillationof the arm about the second axis and the frequency of the oscillation ofthe at least one rotor about the third axis more into phase to increasethe gyroscopic torque.

In some aspects of the present technology, it is expected that the netenergy output from the device will exceed the net energy input into thedevice via the control system. In some aspects of the presenttechnology, the device can provide an energy output that is moreefficient than conventional motor assemblies-even if the energy outputfrom the device is not greater than the energy input to the device.Specifically, the control system can include one or more relativelysmall motor assemblies configured to drive the shaft, the arm, and/orthe rotors. The power input to each of the motor assemblies can berelatively small compared to the total power output of the device viathe shaft. Such smaller motors can be relatively more efficient than acomparable motor assembly configured to directly rotate the shaft toachieve the same output power. Therefore, the arrangement of the devicecan advantageously allow for the power inputs from several smaller motorassemblies to drive a series of motions (e.g., oscillations androtations) that efficiently combine to generate a relatively largerpower output.

Certain details are set forth in the following description and in FIGS.1-16F to provide a thorough understanding of various embodiments of thepresent technology. In other instances, well-known structures,materials, operations, and/or systems often associated with gyroscopes,oscillating and rotating systems, generators, motors, pivotablecouplings, and the like, are not shown or described in detail in thefollowing disclosure to avoid unnecessarily obscuring the description ofthe various embodiments of the technology. Those of ordinary skill inthe art will recognize, however, that the present technology can bepracticed without one or more of the details set forth herein, and/orwith other structures, methods, components, and so forth. Theterminology used below is to be interpreted in its broadest reasonablemanner, even though it is being used in conjunction with a detaileddescription of certain examples of embodiments of the technology.

The accompanying Figures depict embodiments of the present technologyand are not intended to limit its scope unless expressly indicated. Thesizes of various depicted elements are not necessarily drawn to scale,and these various elements may be enlarged to improve legibility.Component details may be abstracted in the Figures to exclude detailssuch as position of components and certain precise connections betweensuch components when such details are unnecessary for a completeunderstanding of how to make and use the present technology. Many of thedetails, dimensions, angles and other features shown in the Figures aremerely illustrative of particular embodiments of the disclosure.Accordingly, other embodiments can have other details, dimensions,angles and features without departing from the present technology. Inaddition, those of ordinary skill in the art will appreciate thatfurther embodiments of the present technology can be practiced withoutseveral of the details described below.

To the extent any materials incorporated herein by reference conflictwith the present disclosure, the present disclosure controls. Theheadings provided herein are for convenience only and should not beconstrued as limiting the subject matter disclosed.

I. Select Embodiments of Centrifugal Gyroscopic Devices and AssociatedSystems and Methods

FIGS. 1A and 1B are partially-schematic isometric views of a centrifugalgyroscopic device 100 (“device 100”) configured in accordance withembodiments of the present technology. The view in FIG. 1B is rotated byabout 90 degrees relative to the view shown in FIG. 1A. FIG. 1C is aside-cross sectional view of the device 100 take along a plane extendingalong a first axis A₁ and a second axis A₂ of the device 100 inaccordance with embodiments of the present technology. Referring toFIGS. 1A-1C together, in the illustrated embodiment the device 100includes a frame 102 comprising (i) a plurality of legs 104 (e.g., fourlegs), (ii) a lower support 106 coupled to or integrally formed with thelegs 104 (e.g., upper portions thereof), and (iii) an upper support 108rotatably coupled to the lower support 106 and the legs 104 via a firstshaft 120 (best seen in FIG. 1C; e.g., a spindle, a drive shaft, anoutput shaft, and/or an elongate member). The upper support 108 isrotatable about the first axis A₁ (e.g., a spindle axis, a verticalaxis, and/or an output axis). In some embodiments, the device 100 can beoriented such that the first axis A₁ extends generally parallel to asurrounding gravitational field. The legs 104 can be fixedly secured(e.g., via one or more fasteners) to the ground, a floor, and/or anothersurface. In some embodiments, the frame 102 further includes a plate 109secured between the legs 104.

In the illustrated embodiment, the device 100 further includes an armassembly 110 pivotably mounted to the upper support 108 within anopening 107 therein via a second shaft 126 (FIG. 1C) extending along thesecond axis A₂. The arm assembly 110 can also be referred to as an arm,a torque-summing assembly, and/or the like. The arm assembly 110includes (i) a housing 112 containing one or more motor assemblies asdescribed in detail below with reference to FIG. 4 , and (ii) a pair ofrotors 114 (which can also be referred to as masses; identifiedindividually as a first rotor 114 a and a second rotor 114 b) pivotablymounted to the housing 112. More specifically, the first rotor 114 a canbe pivotably coupled to/at a first end portion 113 a of the housing 112and the second rotor 114 b can be pivotably coupled to/at a second endportion 113 b of the housing 112 opposite the first end portion 113 a.The housing 112 further includes a central portion 113 c between thefirst and second end portions 113 a-b that is pivotably coupled to thehousing 112 via the second shaft 126 such that the arm assembly 110 ispivotable about the second axis A₂ (e.g., an arm axis, and/or a hingeaxis) orthogonal to the first axis A₁. The rotors 114 are eachindependently pivotable about a third axis A₃ (e.g., a rotor axis,and/or a momentum reference axis) orthogonal to the first axis A₁ andthe second axis A₂ in its reference state. In some embodiments, thelower support 106 and/or the upper support 108 can be omitted and/or thefirst shaft 120 can be directly coupled to the arm assembly 110.

As best seen in FIG. 1C, the first shaft 120 extends through an interiorof the lower support 106 along the first axis A₁ and includes a lowerend portion 121 a and an upper end portion 121 b. In some embodiments,the lower support 106 can at least partially support the first shaft 120via one or more bearings (not shown). The upper end portion 121 b of thefirst shaft 120 is coupled to the upper support 108. Referring to FIGS.1A-1C together, the device 100 includes a shaft motor assembly 130coupled to the plate 109 and operably coupled to the lower end portion121 a of the first shaft 120. The shaft motor assembly 130 can include amotor (e.g., a rotary motor) and associated gearbox and is configured todrive the first shaft 120 to rotate about the first axis A₁ in aclockwise and/or counterclockwise direction as indicated by arrow C inFIG. 1A to drive the upper support 108 and arm assembly 110 to rotatetogether about the first axis A₁. Although the first shaft 120 is shownas spaced apart from the shaft motor assembly 130 in FIGS. 1A-1C forclarity, these components can be directly attached together and/oroperably coupled together via an intervening structure such as a link,coupling, shaft, and/or the like.

In some embodiments, the device 100 can further include a brakemechanism 122 operably coupled to the first shaft 120. The brakemechanism 122 can include a brake plate 123 fixed to the first shaft 120and a brake actuator 124 configured to selectively engage the brakeplate 123 to slow or stop a rotational rate of the first shaft 120. Insome embodiments, the brake mechanism 122 can include other componentsfor selectively slowing the rotational rate of the first shaft 120. Insome embodiments, the brake mechanism 122 can be omitted and/or theshaft motor assembly 130 can be configured to brake/slow rotation of thefirst shaft 120.

In the illustrated embodiment, the device 100 further includes a pair ofarm motor assemblies 132 (individually identified as a first arm motorassembly 132 a and a second arm motor assembly 132 b) coupled to theupper support 108 and operably coupled to the arm assembly 110. FIG. 2is an isometric view of the arm assembly 110 and arm motor assemblies132, configured in accordance with representative embodiments of thepresent technology. Referring to FIGS. 1C and 2 together, morespecifically, the second shaft 126 can include a first end portion 127 aoperably coupled to the first arm motor assembly 132 a and a second endportion 127 b operably coupled to the second arm motor assembly 132 b.The arm motor assemblies 132 can include a motor (e.g., a rotary motor)and associated gearbox and are configured to drive the second shaft 126(FIG. 1C) to rotate about the second axis A₂ to drive the arm assembly110 to pivot about the second axis A₂ in a first direction and/or asecond direction as indicated by arrows F₁ and F₂, respectively, inFIGS. 1A and 2 . In some embodiments, the arm assembly 110 isconstrained (e.g., via one or more mechanical means) to pivot about thesecond axis A₂ in the direction of arrows F₁ and F₂ by a selected (e.g.,predetermined) maximum amplitude. In some embodiments, the arm motorassemblies 132 are identical and are positioned and oriented to besymmetric about the first axis A₁. Accordingly, the arm motor assemblies132 can together define a center of mass positioned along orsubstantially along the first axis A₁. In some embodiments, the device100 can include only a single one of the arm motor assemblies 132 ormore than two of the arm motor assemblies 132.

FIG. 3 is an isometric side view of the arm assembly 110 with thehousing 112 (FIGS. 1A-1C) omitted for clarity in accordance withembodiments of the present technology. In the illustrated embodiment,the arm assembly 110 includes a pair of rotor motor assemblies 334(individually identified as a first rotor motor assembly 334 a and asecond rotor motor assembly 334 b) secured within the housing 112 onopposing sides of the second shaft 126. The first and second rotor motorassemblies 334 a-b can each include a motor (e.g., a rotary motor) andassociated gearbox and are configured to drive the first rotor 114 a andthe second rotor 114 b, respectively, to pivot about the third axis A₃in a first direction and/or a second direction as indicated by arrows G₁and G₂, respectively, in FIG. 1A. In some embodiments, the rotors 114are constrained (e.g., via one or more mechanical means) to pivot aboutthe third axis A₃ in the direction of arrows G₁ and G₂ by a selected(e.g., predetermined) amplitude. In some embodiments, the rotor motorassemblies 334 are identical and are positioned and oriented to besymmetric about the first axis A₁ and the second axis A₂. Accordingly,the rotor motor assemblies 334 can together define a center of masspositioned along or substantially along the second axis A₂. In someembodiments, the device 100 can include only a single one of the rotormotor assemblies 334 or more than two of the rotor motor assemblies 334.

With continued reference to FIG. 3 , in some embodiments the rotors 114can be identical—for example, having the same shape, mass, density,geometry, and/or the like. In some embodiments, the rotors 114 can havea wheel-and-spoke shape such that they are each symmetric about thethird axis A₃. More specifically, the rotors 114 can each include acentral portion 335 coupled to the corresponding one of the rotor motorassemblies 334, an outer portion 336 (e.g., an outer ring), and aplurality of spokes 337 extending radially outward from the centralportion 335 and the third axis A₃ and connecting the central portion 335to the outer portion 336. In some embodiments, the outer portion 336 canhave a greater mass than the central portion 335 such that a majority ofthe mass of the rotor 114 is positioned radially outward from the thirdaxis A₃. In some embodiments, the rotors 114 can have different shapes,sizes, and/or configurations, and/or one of the rotors 114 can beomitted. For example, in some embodiments each of the rotors 114 canhave a planar disc-shape. In some embodiments, the rotors 114 are eachpositioned at a distance D from the second shaft 126 and the second axisA₂ and each have a radius R. The distance D, the radius R, and/or theshape of the rotors 114 can be selected to change the inertia and/ortorque-summing properties of the arm assembly 110 as described ingreater detail below. In some embodiments, the device 100 includes onlyone of the rotors 114 such that the arm assembly 110 is pendulouslyarranged.

Referring to FIGS. 1A and 1B together, the device 100 can furtherinclude a control and power subsystem 140 that in turn includes one ormore power sources 142, one or more sensors 144, one or more powergenerators 146, and a controller 148. The control and power subsystem140 can be operably coupled to the shaft motor assembly 130, the armmotor assemblies 132, the rotor motor assemblies 334 (FIG. 3 ), thebrake mechanism 122, and/or other components of the device 100 via wiredand/or wireless connections. For example, in the illustrated embodimentthe shaft motor assembly 130, the arm motor assemblies 132, and therotor motor assemblies 334 each have one or more electrical connectors141 that can be electrically coupled to the control and power subsystem140 for passing data, power, and/or other signals therebetween.

The power source 142 can be an AC power source and/or a DC power sourceand, in some embodiments, can include/comprise a servo drive 143 (FIG.1B) coupled to the upper support 108 or elsewhere. The power source 142can provide electrical power to the shaft motor assembly 130, the armmotor assemblies 132, and the rotor motor assemblies 334. The sensors144 can include one or more sensors configured (e.g., positioned) fordetecting (i) a rotational and/or pivotal rate of the first shaft 120,the arm assembly 110, and/or the rotors 114, (ii) a power usage of theshaft motor assembly 130, the arm motor assemblies 132, and/or the rotormotor assemblies 334, and/or (iii) a power output of the device 100(e.g., via the power generator 146), and/or the like. The powergenerator 146 can be or can include a dynamo or other suitable generatorcoupled to the first shaft 120 for converting the mechanical rotation ofthe first shaft 120 to electrical energy.

In some embodiments, the power generator 146 can comprise/include theshaft motor assembly 130. That is, the shaft motor assembly 130 candrive the first shaft 120 to rotate in a first operating configurationand convert the rotation of the first shaft 120 to electrical energy ina second operating configuration. In some embodiments, the powergenerator 146 can be or can include a mechanical device for convertingthe mechanical rotation of the first shaft 120 to another useful output.In some embodiments, the power generator 146 can generate electricalenergy and provide the electrical energy to the power source 142 ordirectly to the shaft motor assembly 130, the arm motor assemblies 132,and/or the rotor motor assemblies 334. That is, the power generator 146can function as the power source 142 and/or can provide feedback to thepower source 142. In other embodiments, the power generator 146 cancomprise/include one or both of the arm motor assemblies 132. That is,the arm motor assemblies 132 can drive the second shaft 126 to pivot ina first operating configuration and convert the pivotable motion of thesecond shaft 126 to electrical energy (and/or another useful output,such as mechanical energy) in a second operating configuration.

The controller 148 can receive data from the sensors 144 and control thepower source 142 to operate the shaft motor assembly 130, the arm motorassemblies 132, and/or the rotor motor assemblies 334. Specifically, asdescribed in further detail below, the controller 148 can cause (i) theshaft motor assembly 130 to rotate the first shaft 120 at a selected(e.g., predetermined) rotational rate about the first axis A₁, (ii) thearm motor assemblies 132 to pivot the arm assembly 110 about the secondaxis A₂ at a selected amplitude and frequency, and (iii) the rotor motorassemblies 334 to pivot the rotors 114 about the third axis A₃ at aselected amplitude and frequency. In some embodiments, the controller148 can include/comprise a printed circuit board (PCB) 149 (FIG. 1B)coupled to the upper support 108 or elsewhere.

The controller 148 can comprise a processor and a non-transitorycomputer-readable storage medium that stores instructions that, whenexecuted by the processor, carry out the functions attributed to thecontroller 148 as described herein. Although not required, aspects andembodiments of the present technology can be described in the generalcontext of computer-executable instructions, such as routines executedby a general-purpose computer (e.g., a server or personal computer).Those skilled in the relevant art will appreciate that the presenttechnology can be practiced with other computer system configurations,including Internet appliances, hand-held devices, wearable computers,cellular or mobile phones, multi-processor systems, microprocessor-basedor programmable consumer electronics, set-top boxes, network PCs,mini-computers, mainframe computers and the like. The present technologycan be embodied in a special purpose computer or data processor that isspecifically programmed, configured and/or constructed to perform one ormore of the computer-executable instructions explained in detail below.Indeed, the terms “controller” and “computer” (and like terms), as usedgenerally herein, refers to any of the above devices, as well as anysuitable data processor or any suitable device capable of communicatingwith a network, including consumer electronic goods or other electronicdevices having a processor and other components (e.g., networkcommunication circuitry).

The present technology can also be practiced in distributed computingenvironments, where tasks or modules are performed by remote processingdevices, which are linked through a communications network, such as aLocal Area Network (“LAN”), Wide Area Network (“WAN”), or the Internet.In a distributed computing environment, program modules or sub-routinescan be located in both local and remote memory storage devices. Aspectsof the present technology described below can be stored or distributedon computer-readable media, including magnetic and optically readableand removable computer discs, stored as in chips (e.g., EEPROM or flashmemory chips). Alternatively, aspects of the present technology can bedistributed electronically over the Internet or over other networks(including wireless networks). Those skilled in the relevant art willrecognize that portions of the present technology can reside on a servercomputer, while corresponding portions reside on a client computer. Datastructures and transmission of data particular to aspects of the presenttechnology are also encompassed within the scope of the presenttechnology.

In some embodiments, the control and power subsystem 140, the shaftmotor assembly 130, the arm motor assemblies 132, the rotor motorassemblies 334 can together be referred to as a “control system” or thelike for controlling the motion of the first shaft 120, the arm assembly110, and the rotors 114. Referring to FIGS. 1A-3 together, in generalduring operation of the device 100, the control system is configured tocontrol a rotation rate of the first shaft 120 about the first axis A₁,a frequency and/or amplitude of oscillation of the arm assembly 110about the second axis A₂, and a frequency and/or amplitude ofoscillation of the rotors 114 about the third axis A₃. Morespecifically, the shaft motor assembly 130 can direct the first shaft120 and the coupled arm assembly 110 to rotate about the first axis A₁at and/or to a selected rotational rate. The rotating arm assembly 110generates a centrifugal force that acts against the arm assembly 110 tooscillate the arm assembly 110 about the second axis A₂ in thedirections indicated by arrows F₁ and F₂ in FIG. 1A. In someembodiments, the arm motor assemblies 132 can drive the arm assembly 110about the second axis A₂ (augmenting the centrifugal force actingagainst the arm assembly 110) such that the arm assembly 110 oscillatesat a selected frequency and maximum amplitude. The resulting motion ofthe arm assembly 110 can be periodic (e.g., sinusoidal). The maximumangular amplitude is limited by the torque due to the centrifugal forcefrom the rotation of the arm assembly 110.

The rotor motor assemblies 334 can drive the rotors 114 to pivot aboutthe third axis A₃ in an oscillatory/modulated manner in which the rotors114 pivot in the directions indicated by arrows G₁ and G₂ (FIG. 1A)about the third axis A₃ to a selected frequency and maximum angularamplitude. The resulting motion of the rotors 114 can be periodic (e.g.,sinusoidal). In some embodiments, the periodic motion of the first rotor114 a is opposite (e.g., 180 degrees out of phase with) the periodicmotion of the second rotor 114 b and at the same frequency such that therotors 114 move in opposite directions past one another and reach theirmaximum angular amplitudes in opposite directions at the same time or atleast approximately the same time.

In some embodiments, the power source 142 can supply anoscillatory/modulated voltage to the arm motor assemblies 132 and therotor motor assemblies 334 to generate the oscillatory motion of the armassembly 110 and the rotors 114. The oscillatory angular momentum of therotors 114 and the arm assembly 110 together exert a gyroscopic torqueon the first shaft 120 about the first axis A₁ that acts to rotate thefirst shaft 120. Accordingly, the rotation of the first shaft 120 drivesthe arm assembly 110 to oscillate about the second axis A₂, and theresulting oscillation of the arm assembly 110 and the oscillation of therotors 114 acts to drive the rotation of the first shaft 120 in afeed-back loop. The mechanical rotation of the first shaft 120 can becoupled to the power generator 146 for generating output power.

FIG. 4 is block diagram more specifically illustrating representativephysical properties/forces as the device 100 operates in accordance withembodiments of the present technology. With additional reference toFIGS. 1A-3 , in the illustrated embodiment, the rotor motor assemblies334 provide a drive force 450 that oscillates the rotors 114 about thethird axis A₃. The oscillation of the rotors 114 generates anoscillatory rotor angular momentum 451. The oscillation of the armassembly 110 about the second axis A₂ generates an oscillatory armassembly angular velocity 452. The oscillatory rotor angular momentum451 and the oscillatory arm assembly angular velocity 452 generate agyroscopic torque 454 about the first axis A₁ defined as the vectorcross-product 453 thereof. The gyroscopic torque 454 has an oscillatingmagnitude and average value, which drives the first shaft 120 to rotateat a rotational rate 455. The rotation of the first shaft 120 candeliver mechanical energy to an output load 459, such as the powergenerator 146. The rotation of the first shaft 120 also rotates the armassembly 110, thereby generating a centrifugal force 456 on the armassembly 110 that acts to oscillate the arm assembly 110 about thesecond axis A₂ and generate the oscillatory arm assembly angularvelocity 452. Torque generated on the arm assembly 110 by thecentrifugal force 456 reaches a maximum value when the arm assembly isoriented at a 45 degree angle relative to the second axis A₂ (e.g.,above or below horizontal) and is always directed toward the horizontal.As such, the torque due to the centrifugal force 456 can (i) be theprimary force that drives the arm assembly 110 to have the oscillatoryarm angular velocity 452 and (ii) limit the angular excursion of the armassembly 110 about the second axis A₂ (e.g., with a maximum angularexcursion typically of about 45 degrees).

In some embodiments, the shaft motor assembly 130 can provide anauxiliary drive force 457 that augments the gyroscopic torque 454 tocontrol the rotational rate 455 of the first shaft 120. In someembodiments, the shaft motor assembly 130 can provide the inputauxiliary drive force 457 initially during startup of the device 100until the first shaft 120 is rotating at or near a desired rotationalrate. Likewise, in some embodiments the arm motor assemblies 132 canprovide an auxiliary drive force 458 that oscillates the arm assembly110 to control the oscillatory arm assembly angular velocity 452.

In some embodiments, the device 100 is configured to operate in a“resonant” mode or at least approximately resonant mode (e.g., within atleast 1%, 2%, 5%, 8%, or 10% of resonance) in which the gyroscopictorque 454 is maximized or approximately maximized. In some embodiments,in the resonant mode, the rotational rate of the first shaft 120 (andthe arm assembly 110) about the first axis A₁ equals or at leastapproximately equals an oscillation frequency of the arm assembly 110about the second axis A₂ and an oscillation of frequency of the rotors114 about the third axis A₃. For example, if the first shaft 120 rotatesat 18 rotations per second, the arm assembly 110 and the rotors 114 caneach have an oscillation frequency of 18 hertz in the resonant mode. Insome embodiments, in the resonant mode, the rotational rate of the firstshaft 120 is different (e.g., slightly different) than the oscillationfrequency of the arm assembly 110 and/or the oscillation frequency ofthe rotors 114. That is, there can be a slight dissonance in therotational rate of the first shaft 120, the oscillation frequency of thearm assembly 110, and/or the oscillation frequency of the rotors 114.For example, if the arm assembly 110 and the rotors 114 each have anoscillation frequency of 18 hertz, the first shaft 120 can rotate atbetween about 15-21 rotations per second, and at a rotation ratedifferent than 18 rotations per second, in the resonant mode in whichthe gyroscopic torque 454 is maximized or approximately maximized. Inother embodiments, the device 100 can operate in a “non-resonant” modeof operation in which the rotational rate of the first shaft 120 (andthe arm assembly 110) about the first axis A₁ does not equal theoscillation frequency of the arm assembly 110 about the second axis A₂and the oscillation of frequency of the rotors 114 about the third axisA₃.

Additionally, the device 100 can drive the arm assembly 110 and/or therotors 114 to adjust a phase angle and/or phase relationship between theoscillations of the rotors 114 and the arm assembly 110—and thus a phaseangle between the associated oscillatory rotor angular momentum 451 andthe associated oscillatory arm assembly angular velocity 452. FIGS. 5Aand 5B are graphs illustrating components of the oscillatory rotorangular momentum 451 and the oscillatory arm assembly angular velocity452 of FIG. 4 over time in accordance with embodiments of the presenttechnology. Referring to FIGS. 1A-5B together, the oscillatory rotorangular momentum 451 and the oscillatory arm assembly angular velocity452 can each be periodic due to the oscillatory motion of the rotors 114and the arm assembly 110, respectively. The oscillatory rotor angularmomentum 451 can have a maximum amplitude M_(max) and the oscillatoryarm assembly angular velocity 452 can have a maximum amplitude V_(max).In FIGS. 5A and 5B the oscillatory rotor angular momentum 451 and theoscillatory arm assembly angular velocity 452 have the same frequency(e.g., as in the resonant mode of operation). The oscillatory rotorangular momentum 451 and the oscillatory arm assembly angular velocity452 are offset by a phase angle (in FIG. 5A and are in phase (e.g.,(Φ=0) in FIG. 5B.

In some embodiments, the device 100 is configured to adjust the phaseangle Φ (and/or another phase relationship) between the oscillatoryrotor angular momentum 451 and the oscillatory arm assembly angularvelocity 452 (and the corresponding motion of the rotors 114 and the armassembly 110) to an optimum value such that the device 100 operates inthe resonant or substantially resonant mode and/or maximizes thegyroscopic torque 454 applied to the first shaft 120. To effect such achange in the phase angle Φ, the controller 148 can control the armmotor assemblies 132 and/or the rotor motor assemblies 334 to adjust theoscillations of the arm assembly 110 and/or the rotors 114,respectively. Because the oscillatory rotor angular momentum 451 and theoscillatory arm assembly angular velocity 452 are each sinusoidal (orquasi-sinusoidal), the vector cross-product 453 between them will yieldan average value of the gyroscopic torque 454 and a second harmonic. Theaverage value of the gyroscopic torque 454 is dependent on the phaseangle Φ, and is zero when the oscillatory rotor angular momentum 451 andthe oscillatory arm assembly angular velocity 452 are in quadrature.

In some embodiments, the average value of the gyroscopic torque 454 ismaximized when the phase angle (is 0 degrees or 180 degrees. However,the optimum phase angle Φ that maximizes the value of the gyroscopictorque 454 can have values other than 0 degrees or 180 degrees based onthe operating conditions of the device 100. In particular, the device100 may generate other torques on the arm assembly 110 and/or the rotors114 that inhibit or even prevent the oscillations of the arm assembly110 and the rotors 114 from being totally in phase—that is, such thatthe arm assembly 110 and the rotors have an oscillation component thatis always out of phase.

FIG. 6 is a flow diagram of a method or process 650 for operatingrepresentative devices to generate power in accordance with embodimentsof the present technology. Although some features of the method 650 aredescribed in the context of the device 100 described in detail withreference to FIGS. 1A-5B for the sake of illustration, one skilled inthe art will readily understand that the method 650 can be carried outusing other suitable devices and/or systems described herein.

Beginning at block 651, the method 650 includes rotating the first shaft120 (e.g., a drive shaft) and the arm assembly 110 attached theretoabout the first axis A₁ at a selected rotational rate. In someembodiments, the shaft motor assembly 130 can provide an initial(start-up) torque to rotate the first shaft 120 until the arm assembly110 generates the gyroscopic torque 454, at which point the torque fromthe shaft motor assembly 130 can be reduced or eliminated and therotation of the first shaft 120 driven entirely or substantiallyentirely by the generated gyroscopic torque 454. In some embodiments,the shaft motor assembly 130 can be omitted and instead rotated entirelyby the generated gyroscopic torque 454.

At block 652, the method 650 includes pivoting the arm assembly 110 inan oscillatory manner about the second axis A₂ at a first frequency. Insome embodiments, the arm motor assemblies 132 can provide an initial(start-up) torque to rotate the arm assembly 110 until the centrifugalforce 456 acts to oscillate the arm assembly 110—at which point thetorque from the arm motor assemblies 132 can be reduced or eliminatedand the oscillation of the arm assembly 110 driven entirely orsubstantially entirely by the generated centrifugal force 456. In someembodiments, the arm motor assemblies 132 can be used only to adjust thefrequency of the oscillation of the arm assembly 110 (and a resultingphase relationship between the generated oscillatory arm assemblyangular velocity 452 and the oscillatory rotor angular momentum 451.

At block 653, the method 650 includes pivoting the rotors 114 in anoscillatory manner about the third axis A₃ at a second frequency. Asdescribed in detail above, when the first shaft 120 is rotating (block651), the arm assembly 110 is oscillating (block 652), and (the rotors114 are oscillating (block 653), the device 100 includes a feedback loopin which (i) the angular velocity 452 of the arm assembly 100 and theangular momentum 451 of the rotors 114 combine to generate a gyroscopictorque 454 that acts to rotate the first shaft 120, and (ii) therotation of the first shaft 120 rotates the arm assembly 110 to generatea centrifugal force 456 that acts to oscillate the arm assembly 110.

At block 654, the method 650 includes applying the output load 459 tothe first shaft 120. The output load 459 extracts energy from the device100—acting to slow the rotational rate of the first shaft 120 absent anyadjustments to the operating parameters of the device 100.

At block 655, the method 650 includes controlling the device 100 tooperate in the resonant mode in which the rotational rate of the firstshaft 120 equals or is at least approximately equal to both the firstfrequency of oscillation of the arm assembly 110 and the secondfrequency of oscillation of the rotors 114. In some embodiments,controlling the device 100 to operate in the resonant mode includessetting/adjusting the phase angle (to increase the gyroscopic torque 454applied to the first shaft 120 to compensate for the load 459 andmaintain the resonant mode of operation. Accordingly, in some aspects ofthe present technology the phase angle (can control the resonance of thedevice 100. In some embodiments, the device 100 can be specificallydesigned for a known load such that the phase angle (need not beadjusted/set during operation. However, in some embodiments the load 459can be variable and the oscillations of the arm assembly 110 and/orrotors 114 can be controlled automatically by the controller 148 ormanually by a user to maintain the device 100 in the resonant modeduring variations in the load 459. In other embodiments, controlling thedevice 100 to operate in the resonant mode includes controlling thedevice 100 such that the rotational rate of the first shaft 120 isdifferent (e.g., slightly different) than the first frequency ofoscillation of the arm assembly 110 and/or the second frequency ofoscillation of the rotors 114. That is, the device 100 can be operatedwith a slight dissonance in the rotational rate of the first shaft 120,the first frequency of oscillation of the arm assembly 110, and/or thesecond frequency of oscillation of the rotors 114 to promote/create acentrifugal resonance in which gyroscopic torques are maximized.

In some aspects of the present technology, it is expected that the netenergy output from the device 100 (e.g., via the first shaft 120 to theload 459) will exceed the net energy input into the device 100 via, forexample, the shaft motor assembly 130, the arm motor assemblies 132,and/or the rotor motor assemblies 334. That is, for example, the netmechanical power available from the gyroscopic torque 454 can exceed thesum of the input power to the rotor motor assemblies 334 that drives therotors 114 and the input power to the arm motor assemblies 132 thatprovides auxiliary power to the arm assembly 110 for controlling thephase angle Φ—even when considering impediments (e.g., friction and/orother losses) to the various components of the device 100.

Section II below, for example, models the motion of the device 100 withequations. In Section II, the first axis A₁ is referred to as a “spindleaxis,” the second axis A₂ is referred to as a “hinge axis,” the thirdaxis A₃ is referred to as a “rotor axis,” the first shaft 120 isreferred to as a “spindle,” and the arm assembly 110 is referred to asan “arm.” As outlined in Section II, the difference between the outputpower and the total input power of the device 100 is dependent on thephase angle (and, even factoring in impediments to the motion of thefirst shaft 120, the arm assembly 110, and/or the rotors 114, there canbe a range of phase angles for which the output is greater than theinput. For example, as shown in Section II, the device 100 can generatenet power (e.g., output power greater than input power) for phase angles(between about 5-75 degrees and can generate a maximum amount of netpower when the phase angle (is between about 30-45 degrees (e.g., about40 degrees).

In some aspects of the present technology, the device 100 can provide anenergy output via the first shaft 120 that is more efficient thanconventional motor assemblies-even if the power output from the device100 is not greater than the power input to the device 100. Specifically,the power inputs to the shaft motor assembly 130, the arm motorassemblies 132, and/or the rotor motor assemblies 334 can each berelatively small compared to the total power output of the device 100.Such smaller motors can be relatively more efficient than a comparablemotor assembly configured to directly rotate the first shaft 120 at thesame output power. Therefore, the arrangement of the device 100advantageously allows for the power inputs from several smaller motorassemblies to drive a series of motions (e.g., oscillations androtations) that efficiently combine to generate a relatively great poweroutput.

In general, the operating/design parameters of the device 100 can beoptimized to maximize power output (e.g., based on the equationsdetailed in Section II) based on a selected application of the device100. For example, the device 100 can have a small form factor (e.g., forpowering a watch or cell phone), a medium form factor (e.g., forpowering home appliances), a large form factor (e.g., for powering aremote well or lift station), and so on. Depending on the application,the operating/design parameters that can be optimized include: (i) theoscillation frequency of the rotors 114, (ii) the oscillation amplitudeof the rotors 114, (iii) the oscillation frequency of the arm assembly110, (iv) the oscillation amplitude of the arm assembly 110, (v) thenumber of arm assemblies 110 (e.g., including one or more armassemblies), (vi) the number of rotors 114 mounted to each of the armassemblies 110, (vii) the rotation rate of the first shaft 120, (viii)the size and/or mass of any of the components which can directly affectthe inertia, momentum, velocity, and/or forces generated by thecomponents, among others. For example, Sections III and IV below provideexamples of representative design parameters that can be selected for acentrifugal gyroscopic device in accordance with the present technologythat-even when considering impediments—can operate very efficiently oreven to produce more output than input.

FIG. 7 is a perspective side view of a centrifugal gyroscopic device 700(“device 700”) configured in accordance with additional embodiments ofthe present technology. The device 700 can include some features thatare at least generally similar in structure and function, or identicalin structure and function, to the corresponding features of the device100 described in detail above with reference to FIGS. 1A-6 , and canoperate in a generally similar or identical manner to the device 100. Inthe illustrated embodiment, for example, the device 700 includes: (i) adrive shaft 720 rotatable along the first axis A₁, (ii) an arm assembly710 rotatable with the drive shaft 720 and pivotable about the secondaxis A₂, (iii) a pair of rotors 714 (identified individually as a firstrotor 714 a and a second rotor 714 b) pivotable about the third axis A₃,and (iv) a control and power subsystem 740 operable to drive (via one ormore motor assemblies) the arm assembly 710 to oscillate about thesecond axis A₂ (extending into the page in FIG. 7 ) and the rotors 714to oscillate about the third axis A₃ (and/or to drive the drive shaft720 to rotate about the first axis A₁).

In some embodiments, the device 700 is configured to operate in aresonant mode or substantially resonant mode in which the rotationalrate of the drive shaft 720 (and the arm assembly 710) about the firstaxis A₁ equals or substantially equals an oscillation frequency of thearm assembly 710 about the second axis A₂ and an oscillation offrequency of the rotors 714 about the third axis A₃. FIGS. 8A-8D areenlarged perspective views of the device 700 illustrating the movementof the arm assembly 710 and the rotors 714 during one completerevolution of the arm assembly 710 about the first axis A₁ in theresonant mode of operation in accordance with embodiments of the presenttechnology. FIGS. 8A-8D sequentially illustrate the arm assembly 720 atdifferent quarter (e.g., 90 degrees) revolutions about the first axisA₁. If the rotation of the arm assembly 720 is at a sufficient rate toblur the oscillatory movement of the arm assembly 720, an opticalillusion presents itself that the axis of rotation of the arm assembly720 is tilted by the angle shown in FIGS. 8B and 8D.

In FIG. 8A, the arm assembly 710 extends generally parallel to thehorizontal (e.g., horizontal to gravity). That is, the third axis A₃extends orthogonal to the first axis A₁ (e.g., such that an anglebetween the first and third axes A₁, A₃ is about 90 degrees).Accordingly, the arm assembly 710 can have a minimum angular amplitudein the position in shown FIG. 8A. Further, the rotors 714 are each at afirst angular position relative to the third axis A₃.

In FIG. 8B, the arm assembly 710 has rotated by about 90 degrees (e.g.,in a clockwise direction) about the first axis A₁ from the positionshown in FIG. 8A and such that arm assembly 710 extends at angle Trelative to the horizontal. In the illustrated embodiment, the secondrotor 714 b is positioned above the first rotor 714 a relative to thehorizontal. In some embodiments, the angle T can be a maximum angularamplitude of the oscillation of the arm assembly 720. Further, therotors 714 are each at a second angular position relative to the thirdaxis A₃ in FIG. 8B. In some embodiments, where the phase of the armassembly 710 and the rotors 714 is generally the same (e.g., phase angleequal to zero), the rotors 714 can—the same as the arm assembly 720—havea maximum angular amplitude in FIG. 8B and a minimum angular amplitudein FIG. 8A.

In FIG. 8C, the arm assembly 710 has rotated by about 90 degrees (e.g.,in a clockwise direction) about the first axis A₁ from the positionshown in FIG. 8B and such that the arm assembly 710 extends generallyparallel to the horizontal and again has the minimum angular amplitude.The rotors 714 are each at a third angular position relative to thethird axis A₃ in FIG. 8C. In the resonant mode where the oscillationfrequency of the arm assembly 710 equals the oscillation frequency ofthe rotors 714, the third angular position can be the same as the firstangular position (e.g., both at the minimum angular amplitude) or thethird angular position can have the opposite sign as the first angularposition.

In FIG. 8D, the arm assembly 710 has rotated by about 90 degrees (e.g.,in a clockwise direction) about the first axis A₁ from the positionshown in FIG. 8C and such that arm assembly 710 extends again at theangle T relative to the horizontal. In the illustrated embodiment,however, the first rotor 714 a is positioned above the second rotor 714b relative to the horizontal. The rotors 714 are each at a fourthangular position relative to the third axis A₃ in FIG. 8D. In theresonant mode where the oscillation frequency of the arm assembly 710equals the oscillation frequency of the rotors 714, the fourth angularposition can be the same as the second angular position (e.g., both at amaximum angular amplitude) or the fourth angular position can have theopposite sign as the second angular position.

The arm assembly 710 completes a revolution by returning to the positionshown in FIG. 8A. Referring to FIGS. 8A-8D together, the movement of thedevice 700 in the resonant mode can create an optical illusion whenviewed from the side at a particular azimuthal angle as shown in FIGS.8A-8D. Namely, the arm assembly 710 can appear to be consistentlypivoted about the second axis A₂ (FIG. 7 ) off vertical (e.g., typicallyat a tilt angle of 45 degrees) in a particular direction. This occursbecause, for example, the rotors 714 each reach a maximum angularamplitude below horizontal when located at the same or approximately thesame circumferential position about the first axis A₁ (e.g., to the leftof the page as shown in FIGS. 8B and 8D). Similarly, the rotors 714 eachreach a maximum angular amplitude above horizontal when located at thesame or substantially the same circumferential position about the firstaxis A₁ (e.g., to the right of the page as shown in FIGS. 8B and 8D). Insome embodiments, the optical illusion will rotate in azimuth when thedevice 100 is off resonance.

FIG. 9 is a schematic diagram of a control assembly for controlling aspindle (e.g., the first shaft 120) in accordance with embodiments ofthe present technology. FIG. 10 is a schematic diagram of a controlassembly for controlling a spindle (e.g., the first shaft 120) inaccordance with additional embodiments of the present technology. FIG.11 is a schematic diagram of an analog control assembly for controllinga spindle (e.g., the first shaft 120) in accordance with additionalembodiments of the present technology.

II. Additional Equations Representative of Operating CentrifugalGyroscopic Devices

A set of gyroscopic axes is defined by the vector cross-product,according to the righthand rule. The axes are termed the rotor referenceaxis, the spindle axis, and the hinge axis. For convenience, the spindleaxis is considered to be vertical. If the rotor axis is at an angle tothe rotor reference axis about the hinge axis, the vector cross-productautomatically takes it into account. The vector cross-product of (vectoralong the rotor axis)×(vector along the spindle axis) yields a (vectoralong the hinge axis). Similarly, the vector cross-product of (vectoralong the spindle axis)×(vector along the hinge axis) yields a (vectoralong the rotor axis). And, similarly, the vector cross-product of(vector along the hinge axis)×(vector along the rotor axis) yields a(vector along the spindle axis). Reverse vector cross-products are alsoapplicable. Specifically, the vector cross-product of (vector along thespindle axis)×(vector along the rotor axis) yields a (vector along thenegative hinge axis). Similarly, the vector cross-product of (vectoralong the hinge axis)×(vector along the spindle axis) yields a (vectoralong the negative rotor axis). And, similarly, the vector cross-productof (vector along the rotor axis)×(vector along the hinge axis) yields a(vector along the negative spindle axis). The six vector cross-productsare summarized as follows:

τ_(R)×τ_(S)=τ_(H)

τ_(S)×τ_(H)=τ_(R)

τ_(S)×τ_(R)=−τ_(H)

τ_(H)×τ_(S)=−τ_(R)

τ_(R)×τ_(S)=τ_(H)

τ_(S)×τ_(R)=−τ_(H)

where: τ_(R)=unit vector along rotor axis

-   -   τ_(S)=unit vector along spindle axis    -   τ_(H)=unit vector along hinge axis

In terms of gyroscopic action, each vector can represent either angularmomentum or angular velocity. The vector cross-product of angularmomentum and its angular velocity about an orthogonal axis yields agyroscopic torque about the third axis. Thus, in general, the angularmomentum of the rotor, arm, and spindle can each have an angularvelocity about either of the two orthogonal axes, yielding sixcombinations of gyroscopic torque.

In this particular case, the spindle has only one degree of freedom,that is, rotation about the spindle axis. Therefore, its angularmomentum is restricted from rotating about an orthogonal axis, thuseliminating two of the generalized gyroscopic torques, leaving four.

Also, the arm is prohibited from rotating about the rotor axis.Therefore, the gyroscopic torque that would arise from the angularmomentum of the arm being rotated about the rotor axis is eliminated,leaving three. Further, although the spindle rotates the angularmomentum of the arm, thus exerting a gyroscopic torque on the arm aboutthe rotor axis, the arm is prohibited from rotating about the rotoraxis. This gyroscopic torque exerts a stress on the supporting structureof the hinge axis. With the elimination of this gyroscopic torque ashaving an effect on the dynamic motion in the system, there remain twogyroscopic torques that do affect the dynamic motion of the system. Thetwo gyroscopic torques that are effective are as follows:

M _(R/S) =−I _(R/R)ω_(R/R)ω_(R/H) cos θ_(A/H)

M _(R/H) =I _(R/R)ω_(R/S) cos θ_(A/H)

where: I_(R/R)=moment of inertia of rotor about rotor axis

-   -   ω_(R/R)=angular velocity of rotor about rotor axis    -   ω_(R/S)=angular velocity of rotor and arm about spindle axis    -   ω_(R/H)=angular velocity of rotor and arm about hinge axis    -   θ_(A/H)=arm angle off horizontal about hinge axis    -   M_(R/S)=gyroscopic torque on rotor and arm about spindle axis    -   M_(R/H)=gyroscopic torque on rotor and arm about hinge axis

If the angular momentum about one axis and its velocity about anorthogonal axis are both constant, the gyroscopic torque about the thirdaxis is constant. If either the angular momentum or angular velocity isoscillatory and the other is constant, the gyroscopic torque isoscillatory. If both the angular momentum and angular velocity areoscillatory at the same frequency, the gyroscopic torque has twocomponents, a constant (which is zero, if the oscillations are inquadrature, that is, 90 degrees apart in phase) and a second harmonic.

For an arm that is balanced along the rotor reference axis and along thespindle axis, the differential equations for rotor oscillation about therotor axis, spindle rotation, and arm oscillation about the hinge axis,respectively, are given by:

I _(R/R){umlaut over (θ)}_(R/R)=(M _(R))_(IN)−(M _(R))_(I) I_(S/S){umlaut over (θ)}_(S) =−I _(R/R){dot over (θ)}_(R/R){dot over(θ)}_(A/H)+cos θ_(A/H)+(M _(S))_(IN)−(MS)_(I) −M _(LOAD) I _(A/H){umlautover (θ)}_(A/H) =[−m _(A)(r _(G/R) ² −r _(G/S) ²)({dot over (θ)}_(S))²sin θ_(A/H) cos θ_(A/H) +I _(R/R){dot over (θ)}_(R/R){dot over (θ)}_(S)cos θ_(A/H)+(M _(A))_(M)−(M _(A))_(I)]

where: I_(R)=Moment of inertia of rotor about rotor axis

-   -   I_(S/S)=Moment of inertia of spindle (including arm) about        spindle axis    -   I_(A/H)=Moment of inertia of arm about hinge axis    -   {umlaut over (θ)}_(R/R)=Angular acceleration of rotor about        rotor axis    -   {umlaut over (θ)}_(S)=Angular acceleration of spindle about        spindle axis    -   {umlaut over (θ)}_(A/H)=Angular acceleration of arm about hinge        axis    -   {dot over (θ)}_(R/R)=Angular velocity of rotor about rotor axis    -   {dot over (θ)}_(S)=Angular velocity of spindle about spindle        axis    -   {dot over (θ)}_(A/H)=Angular velocity of arm about hinge axis    -   θ_(A/H)=Angle of arm horizontal    -   (M_(R))_(IN)=Input torque for rotor oscillation    -   (M_(S))_(IN)=Input torque for startup of spindle rotation    -   (M_(A))_(M)=Input torque about hinge axis    -   (M_(R))_(I)=Impediment torque to rotor oscillation about rotor        axis    -   (M_(A))_(I)=Impediment torque to arm oscillation about hinge        axis    -   (M_(S))_(I)=Impediment torque to spindle rotation    -   M_(LOAD)=Load torque on spindle    -   m_(A)=Mass of arm    -   r_(G/R)=Radius of gyration along rotor reference axis    -   r_(G/S)=Radius of gyration along spindle axis

The equations are mathematically intractable, mainly because theexpression for the torque due to centrifugal force contains the productof the sine and cosine of the angle of the arm about the hinge axis,which itself is a quasi-sinusoidal function. A closed form solution isavailable under ideal impediment-free conditions and small amplitudes ofoscillation where small-angle approximations are valid.

The spindle velocity is assumed to be constant. The phase of the torquefrom the motor is assumed to be in phase with the rotor angularacceleration. The rotor and arm are oscillated with the same frequencyof oscillation. The differential equation for motion about the hinge/armaxis can be rewritten with slightly different nomenclature, as follows:

I _(A){umlaut over (θ)}_(A) =M _(A) +I _(R){dot over (θ)}_(R)ω_(S) −I_(C)ω_(S) ²θ_(A)

-   -   I_(A)=Moment of inertia of arm and rotor about the arm axis    -   I_(R)=Moment of inertia of rotor about rotor axis    -   M_(A)=Torque provided by arm motor    -   ω_(S)=Angular velocity of spindle    -   I_(C)=Constant relating spindle speed to centrifugal torque on        arm

The arm position is assumed to have the amplitude θ_(A,max) at time t=0.The rotor acceleration and the arm torque have a phase ϕ relative to thearm position.

θ_(A)=θ_(A,max) cos ωt

{dot over (θ)}_(A)=−ωθ_(A,max) sin ωt

{umlaut over (θ)}_(A)=−ω²θ_(A,max) cos ωt

M _(A) =−M _(A,max) cos(ωt+ϕ)

${\overset{¨}{\theta}}_{A} = {\frac{M_{A}}{I_{A}} + {\frac{I_{R}}{I_{A}}{\overset{.}{\theta}}_{R}\omega_{S}} - {\frac{I_{C}}{I_{A}}\omega_{S}^{2}\theta_{A}}}$

-   -   let ω=ω_(R)=ω_(A)=frequency of oscillation    -   let ω_(S)=ω    -   M_(A) and {umlaut over (θ)}_(R) are in phase

θ_(R)=θ_(R,max) cos(ωt+ϕ)

{dot over (θ)}_(R)=−ωθ_(R,max) sin(ωt+ϕ)

{umlaut over (θ)}_(R)=−ω²θ_(R,max) cos(ωt+ϕ)

Substitute these values for rotor and arm positions, velocities, andaccelerations into the torque equation.

${{- \omega^{2}}\theta_{A,\max}\cos\omega t} = {{{- \left( \frac{M_{A,\max}}{I_{A}} \right)}{\cos\left( {{\omega t} + \phi} \right)}} - {\frac{I_{R}}{I_{A}}\omega\omega_{S}\theta_{R,\max}{\sin\left( {{\omega t} + \phi} \right)}} - {\frac{I_{C}}{I_{A}}\omega_{S}^{2}\theta_{A,\max}{\cos\left( {\omega t} \right)}}}$

Replace cos(ωt) with

cos (ωt + ϕ − ϕ) = sin ϕsin(ωt + ϕ) + cos ϕcos (ωt + ϕ).${{{- \omega^{2}}\theta_{A,\max}\sin{{\phi sin}\left( {{\omega t} + \phi} \right)}} - {\omega^{2}\theta_{A,\max}\cos{{\phi cos}\left( {{\omega t} + \phi} \right)}}} = {{{- \left( \frac{M_{A,\max}}{I_{A}} \right)}{\cos\left( {{\omega t} + \phi} \right)}} - {\frac{I_{R}}{I_{A}}\omega\omega_{S}\theta_{R,\max}{\sin\left( {{\omega t} + \phi} \right)}} - {\frac{I_{C}}{I_{A}}\omega_{S}^{2}\theta_{A,\max}\sin\phi{\sin\left( {{\omega t} + \phi} \right)}} - {\frac{I_{C}}{I_{A}}\omega_{S}^{2}\theta_{A,\max}\cos\phi{\cos\left( {{\omega t} + \phi} \right)}}}$

Collecting the sin(ωt+ϕ) terms results in:

${{{- \omega^{2}}\theta_{A,\max}\sin\phi} = {{{- \frac{I_{R}}{I_{A}}}\omega\omega_{S}\theta_{R,\max}} - {\frac{I_{C}}{I_{A}}\omega_{S}^{2}\theta_{A,\max}\sin\phi}}}{\theta_{R,\max} = {\frac{\left( {\omega^{2} - {\frac{I_{C}}{I_{A}}\omega_{S}^{2}}} \right)}{\frac{I_{R}}{I_{A}}\omega\omega_{S}}\theta_{A,\max}\sin\phi}}$

Define

$\omega_{o} = {\sqrt{\frac{I_{C}}{I_{A}}}\omega_{s}}$

Resonance for the arm would occur at

$\omega_{o} = {\sqrt{\frac{I_{C}}{I_{A}}}\omega_{s}}$

which implies that ω_(o)<ω_(S) since I_(C)<I_(A).

${\theta_{R,\max} = {\left( \frac{I_{A}}{I_{R}} \right)\frac{\left( {\omega^{2} - \omega_{o}^{2}} \right)}{\omega\omega_{S}}\theta_{A,\max}\sin\phi}}{\theta_{A,\max} = {\left( \frac{I_{R}}{I_{A}} \right)\frac{\omega\omega_{S}}{\left( {\omega^{2} - \omega_{o}^{2}} \right)\sin\phi}\theta_{R,\max}}}$

There is no resonance for the arm for ω_(S)=ω. True, resonance isindicated when spindle speed is a bit higher than the oscillatingfrequency of the rotors and arm, if the small-angle approximations werestill valid at large amplitudes and with impediment torques acting onthe arm. The stiffness afforded by the centrifugal-forced torque is notconstant at larger amplitudes. Nevertheless, it is an intriguing notionas to whether some resonant-like behavior can be used to advantage atlarger amplitudes. As a cautionary note, as arm amplitudes ofoscillation are enlarged, they become subject to limitations imposed bycentrifugal effects.

There is no indication of resonance for the rotor.

Collecting the cos(ωt+ϕ) terms gives:

${{{- \omega^{2}}\theta_{A,\max}\cos\phi} = {{- \left( \frac{M_{a,\max}}{I_{A}} \right)} - {\frac{I_{C}}{I_{A}}\omega_{S}^{2}\theta_{A,\max}\cos\phi}}}{\frac{M_{A,\max}}{I_{A}} = {\left( {\omega^{2} - \omega_{o}^{2}} \right)\theta_{A,\max}\cos\phi}}{M_{A,\max} = {{I_{A}\left( {\omega^{2} - \omega_{o}^{2}} \right)}\theta_{A,\max}\cos\phi}}{M_{A,\max} = {I_{A}{\omega^{2}\left( {1 - \frac{I_{C}}{I_{A}}} \right)}\theta_{A,\max}\cos\phi}}$

Dividing these two equations:

${\tan\phi} = \frac{\theta_{R,\max}I_{R}\omega\omega_{S}}{M_{A,\max}}$

The gyroscopic torque applied to the spindle is given by this equation:

$M_{POGA} = {{{- I_{R}}{\overset{.}{\theta}}_{R}{\overset{.}{\theta}}_{A}} = {{{- {I_{R}\left( {{- \omega}\theta_{R,\max}{\sin\left( {{\omega t} + \phi} \right)}} \right)}}\left( {{- \omega}\theta_{A,\max}\sin\omega t} \right)} = {{{- I_{R}}\omega^{2}\theta_{R,\max}{\theta_{A,\max}\left( {{{\sin^{2}\left( {\omega t} \right)}\cos\phi} + {{\sin\left( {\omega t} \right)}{\cos\left( {\omega t} \right)}\sin\phi}} \right)}} = {{- I_{R}}\omega^{2}\theta_{R,\max}{\theta_{A,\max}\left( {{\frac{1}{2}\left( {1 - {\sin\left( {2\omega t} \right)}} \right)\cos\phi} + {{\sin\left( {\omega t} \right)}{\cos\left( {\omega t} \right)}\sin\phi}} \right)}}}}}$

Integrate with respect to time to get the average gyroscopic torque. Thesin(2 ωt) and the sin(ωt) terms integrate to zero.

$\overset{\_}{M_{POGA}} = {{- \frac{1}{2}}I_{R}\omega^{2}\theta_{R,\max}\theta_{A,\max}\cos\phi}$

As was mentioned for θ_(A,max) previously, if the solution withsmall-angle approximations is a harbinger of performance in general, theaverage gyroscopic torque can be enlarged by approaching resonance.

Replace θ_(R,max) with expression above from sin terms:

${\overset{\_}{M_{POGA}} = {{- \frac{1}{2}}I_{A}\omega\frac{\left( {\omega^{2} - \omega_{o}^{2}} \right)}{\omega_{S}}\theta_{A,\max}^{2}\sin\phi\cos\phi}}{\overset{\_}{M_{POGA}} = {{- \frac{1}{2}}I_{A}\omega\frac{\left( {\omega^{2} - \omega_{o}^{2}} \right)}{\omega_{S}}\theta_{A,\max}^{2}{\sin\left( {2\phi} \right)} \times 0.5}}{{\overset{\_}{M_{POGA}}\omega_{S}} = {P_{out} = {{- \frac{1}{2}}I_{A}{\omega\left( {\omega^{2} - \omega_{o}^{2}} \right)}\theta_{A,\max}^{2}{\sin\left( {2\phi} \right)} \times 0.5}}}$

When the spindle is used as the source of output power, it is notconvenient to compare the output torque with the input torques, becausethe output is an average value and the input torques are oscillatory.However, the output power can be compared to the total input power. Theoutput power is the product of the average output torque and the spindlerate. Each input power is the average power over a quarter oscillatorycycle (being independent of algebraic signs and being the same in eachof the other three quarters) and is given by the average of the productof the oscillatory input torque and the instantaneous angular velocityattributed to input torque. The expressions for output power and inputpower to the rotor and to the arm by an auxiliary motor are given by:

${P_{OUT} = {{{\overset{\_}{M}}_{OUT}\omega} = {\frac{1}{2}I_{R/R}{\omega^{3}\left( \theta_{A/H} \right)}_{\max}^{2}\left( \frac{r_{G/S}}{r} \right)^{2}\sin\phi\cos\phi}}}\begin{matrix}{P_{{IN},R} = {\frac{1}{\pi/2}{{{\int}_{0}^{2}\left\lbrack {I_{R/R}{\omega^{2}\left( \theta_{R/R} \right)}_{\max}\sin\omega\tau} \right\rbrack}\left\lbrack {{\omega\left( \theta_{R/R} \right)}_{\max}\cos\omega\tau} \right\rbrack}{\phi\left\lbrack {\omega\tau} \right\rbrack}}} \\{= {{\frac{1}{\pi}I_{R/R}{\omega^{3}\left( \theta_{R/R} \right)}_{\max}^{2}} = {\frac{1}{\pi}I_{R/R}{\omega^{3}\left( \frac{r_{G/S}}{r} \right)}^{\tau}\left( \theta_{A/H} \right)_{\max}^{2}\sin^{2}\phi}}}\end{matrix}\begin{matrix}{P_{{IN},M} = {\frac{1}{\pi}{\left( M_{A} \right)_{M,\max}\left\lbrack {\omega\left( \frac{\left( M_{A} \right)_{M,{NR}}}{I_{R/R}\omega^{2}} \right)} \right\rbrack}}} \\{= {\frac{1}{\pi}I_{R/R}{\omega^{3}\left( \frac{r_{G/S}}{r} \right)}^{2}\left( \frac{r_{G/S}}{r_{G/R}} \right)^{2}\left( \theta_{A/H} \right)_{\max}^{2}\cos^{2}\phi}}\end{matrix}$

The difference between the output power and total input power can beexpressed in normalized fashion as follows:

$\frac{P_{OUT} - \left( {P_{{IN},R} + P_{{IR},M}} \right)}{I_{R/R}{\omega^{3}\left( \theta_{A/H} \right)}_{\max}^{2}} = {{\frac{1}{2}\left( \frac{r_{G/S}}{r} \right)^{2}\sin\phi\cos\phi} - {\frac{1}{\pi}\left( \frac{r_{G/S}}{r} \right)\sin^{2}\phi} - {\frac{1}{\pi}\left( \frac{r_{G/S}}{r} \right)^{2}\left( \frac{r_{G/S}}{r_{G/R}} \right)^{2}\cos^{2}\phi}}$

FIG. 12 is a graph illustrating normalized net power versus the phaseangle between rotor and arm oscillations. As shown, the normalized netpower depends on the phase angle between the rotor and arm oscillationsfor a practical set of design geometries. There is a range of phaseangles where the output is greater than the input.

Alternately, the input power is given by the in-phase component of theproduct of torque and velocity. The expressions for the rotor and armmotors are:

P−in−r=2×time integral of(M−r−max*sin(2*pi*f*t))*(w−r−max*sin(2*pi*f*t+phi−r)/T

P−in−a=time integral of(M−a−max*sin(2*pi*f*t))*(w−a−max*sin(2*pi*f*t+phi−a)/T

M−r−max=Kt−r*Amp−r−max

M−a−max=Kt−a*Amp−a−max

where: P−in−r=input power from each rotor motor (watts)

-   -   P−in−a=input power from arm motor (watts)    -   M−r−max=amplitude of torque provided by rotor motor (N−m)    -   M−a−max=amplitude of torque provided by arm motor (N−m)    -   Kt−r=torque constant for rotor motor (N−m/amp)    -   Kt−a=torque constant for arm motor (N−m/amp)    -   Amp−r−max=amplitude of current in rotor motor (amp)    -   Amp−a−max=amplitude of current in arm motor (amp)    -   w−r−max=amplitude of rotor angular velocity about rotor axis        (rad/sec)    -   w−a−max=amplitude of arm angular velocity about arm axis        (rad/sec)    -   f=frequency of oscillation of rotor and arm (hz)    -   phi−r=phase angle between rotor torque and angular velocity        waveforms (rad)    -   phi−a=phase angle between arm torque and angular velocity        waveforms (rad)    -   t=time (sec)    -   T=1/f=period of oscillation of rotor and arm (sec)    -   pi=3.14159

The mechanical efficiency in percent is expressed as:

Eff−mech=P−out/(P−in−r+P−in−a)×100

where: Eff−mech=efficiency of mechanical power in instrument (%)

When the arm is used as the source of output power, it provides AC powerat the oscillation frequency. The input power is supplied by a spindlemotor and the rotor motors. The expression for the spindle motor isgiven by:

P−in−s=M−s*w−s=M−s*(w−s′)*(2*pi/60)

M−s=Kt−s*Amp−s

where: P−in−s=input power from spindle motor (watts)

-   -   M−s=torque provided by spindle motor (N−m)    -   Kt−s=torque constant for spindle motor (N−m/amp)    -   Amp−s=current in spindle motor (amp)    -   w−s=spindle rate (rad/sec)    -   w−s′=spindle speed (rps)

The expression for the rotor motor power to oscillate the rotor is givenby:

P−in−r=2×time integral of(M−r−max*sin(2*pi*f*t))*(w−r−max*sin(2*pi*f*t+phi−r)/T

The input power supplied by the rotor motors to the arm is given by:

P−r/a−max=2×(M−r−max*sin(2*pi*f*t))*(w−a−max*sin(2*pi*f*t+phi−r/a)

where: P−r/a−max=amplitude of power supplied by rotor motor to arm(N−m/sec)

-   -   phi−r/a=phase angle between rotor torque and arm angular        velocity waveforms (rad)

The input power supplied by centrifugal-forced torque to the arm isgiven by:

P−cent−max=(2×I−cent*(w−s){circumflex over( )}2*sin(Theta−a)*cos(Theta−a)*w−a−max*sin(2*pi*f)

Theta−a=Theta−a−max*sin(2*pi*f*t)

where: P−cent−max=amplitude of power supplied centrifugal forces(N−m/sec)

-   -   I−cent=constant relating spindle speed to centrifugal torque on        half-arm (N−m−sec{circumflex over ( )}2)    -   Theta−a=arm angle (rad)    -   Theta−a−max=amplitude of arm oscillation (rad)

The power required to oscillate the arm is given by:

P−a−osc−max=2×(MOI−a*(2*pi*f*t)*Theta−a−max*sin(2*pi*f*t))*(w−a−max*sin(2*pi*f*t)

where: P−a−osc−max=power required to oscillate arm to Theta−a−maxamplitude (watts)

-   -   MOI−a=moment of inertia of half arm (N−m−sec{circumflex over        ( )}2)

The output power available from the oscillation of the arm is thedifference between the total power being applied to the arm and thepower required to oscillate the arm. The expression for the output poweris given by the equations:

P−out=(P−r/a−avg+P−cent−avg)−P−a−osc−avg

where: P−out=mechanical power available (watts)

-   -   P−r/a−avg=contribution of power from rotor motor to oscillate        arm (watts)    -   P−cent−avg=contribution of power from centrifugal forces to        oscillate arm (watts)    -   P−a−osc−avg=power required to oscillate arm to Theta-a-max        amplitude (watts)

The input power supplied by a spindle motor and the rotor motors is evenby:

P−in=P−in−s+P−in−r+P−a−max

where: P−in =total input power supplied to operate instrument (watts)

The mechanical efficiency in percent is given by:

Eff−mech=P−out/(P−in)×100

III. Select Embodiments of Representative Design Parameters andPerformance Calculations of Centrifugal Gyroscopic Devices

Table 1 below provides a list of the design parameters and calculatedperformance for a representative embodiment of the centrifugalgyroscopic device 100 shown in FIGS. 1A-1C.

TABLE 1 Calculated Performance and/or User Input (bold + DesignParameter underline) Units Notes General Device Parameters Rotor Inertia1.20E−03 kg-m² Rotor Radius 0.10 m Rotor Mass 0.12 kg I = mr² Assumesall mass is at radius, ignores motor. Rotor Arm Length 0.18 m Minimum of2 × (Rotor Radius). Rotor Amplitude 12.00   deg Rotor Amplitude(Radians) 0.21 rad Spindle Frequency 4.17 Hz Spindle RPM 250    RPMSpindle Angular 26.20  rad/s Velocity (W) For Sinusoidal Motion RotorPosition Maximum 0.21 rad Max Rotor Angular Velocity (W) 5.49 rad/sec(Max Rotor Position) × (Spindle W). Max Rotor RPM 52    RPM Convert toRPM. Max Rotor Acceleration 144    rad/s² (Max Rotor W) × (Spindle W).Max Rotor Torque 0.17 N-m Motor torque required. (Rotor Inertia) × (MaxRotor Acceleration). Rotor Angular Momentum (L)  0.0132 kg-m²/ (2 ×Angular Momentum) × (Rotor sec² W (both rotors)). Rotor Arm ArmAmplitude 3.60 degrees Max Arm Position 0.06 rad Amplitude converted toradians. Max Arm Angular Velocity (W) 1.65 rad/sec (Amplitude) ×(Spindle W). Arm Acceleration 43.13  rad/sec² (Max Arm W) × (Spindle W).Arm Inertia from 7.52E−03 kg-m² Both rotors (2 m × (Rotor Arm RotorsLength)²) Torques Max Torque from Rotors 0.32 N-m Motor torque requiredto tilt the arm due to acceleration considering rotors only. EstimatedInertia 0.07 kg-m² 4 kg at 0.1 m. Not Including Rotors Estimated Torque3.02 N-m Not Including Rotors Max Total Torque 3.34 N-m Total motortorque required to tilt arm (not considering the torque that the rotorsand spindle apply to the arm). Arm Moment of Inertia 0.08 kg-m² Rotorsplus estimate of rest of arm. Arm Angular Momentum 2.03 kg-m²/ Angularmomentum around spindle. sec² Centrifugal Forces Force at Horizontal14.58  N On one rotor m × w² × r Torque from Both Rotors 2.58 N-m Forceat horizontal times sin(2*theta) from 2*cos(theta)*sin(theta). Torque at45 degrees. This torque is direction of rotor arm axis. This torque isin a direction to restore the arm to horizontal. Max Gyroscopic Torque0.35 N-m (Rotor Angular Momentum) × on Arm (Spindle W). Max torque onarm from rotor momentum at horizontal. This torque opposes torque thatthe arm is applying to the rotor. Max Gyroscopic Torque 0.02 N-m (RotorAngular Momentum) × (Arm on Spindle W). Max torque on spindle from rotormomentum due to arm movement. Powers Spindle Power 0.57 watts Maxinstantaneous power @600 RPM = (Gyroscopic torque) × (Spindle W due toarm movement). Rotor Arm Power 5.50 watts Max instantaneous rotor armpower = (Max motor torque required) × (Max Rotor Arm W) Rotor Power 0.95watts Max instantaneous rotor power = (Max Rotor Torque) × (Max Rotor W)Other Performance Calculations Rotor Inertia Diameter 6.00E−04 kg-m²Moment of inertia of rotor about diameter. Rotor Arm Torque 0.01 N-mTorque on rotor at max arm velocity. Rotor Spindle Torque 0.03 N-mTorque on rotor at spindle velocity. Centrifugal Force Rotor 15.22  NThrust force on rotor motor. Rotor Acceleration Torque 0.03 N-m Torqueon rotor shaft by rotor accelerating by the arm movement.

FIG. 13A is a graph illustrating sample test results for the centrifugalgyroscopic device 100 having the characteristics described in Table 1above showing the change in mechanical power of the spindle versus theamplitude of oscillation of the rotors. Table 2 below providescorresponding sample test results showing the change in mechanical powerof the spindle versus the amplitude of oscillation of the rotors. FIG.13B is a graph illustrating sample test results for the centrifugalgyroscopic device 100 having the characteristics described in Table 1showing the extraction of gyroscopic power from spindle rate harmonics.The data shown in FIGS. 13A and 13B was computed from the measurement oftwo-cycle torque in the spindle using discrete Fourier transform (DFT)analysis. To produce the test results shown in FIGS. 13A, 13B, and Table2, the pair of rotors were oscillated in synchrony with each other. Thearm oscillated the rotors about an orthogonal axis. The oscillation ofthe arm was in synchrony with the rotor oscillation. The oscillationsgenerated output gyroscopic torque about the spindle axis. In thetesting, the gyroscopic torque was not large enough to overcome thelarge impediment torques on the spindle, thus requiring aid from thespindle motor. The rotating spindle feeds back gyroscopic andcentrifugal torques to amplify the amplitude of arm oscillation. Spindlerate was adjusted for centrical resonance. The test data shows that forcertain regions of operation mechanical power output can be greater thanthe input (e.g., with the rotor amplitude at 18 degrees, 27 degrees, or36 degrees).

TABLE 2 Output Power Rotor Change in Gyroscopic Rotor Arm Total MinusAmplitude Spindle Power Power Power Power Input Power Input Power(Degrees) (Watts) (Watts) (Watts) (Watts) (Watts) (Watts) 9 0*    0** 0.22 −0.13 0.09 −0.09 (Reference (Gyroscopic power Value) value unknown)18 −0.58 0.58 0.80 −0.49 0.31 0.27 27 −0.82 0.82 1.24 −0.87 0.37 0.45 36−1.87 1.87 1.42 −1.22 0.20 1.67

FIGS. 14A-14C are graphs illustrating further sample test results forthe centrifugal gyroscopic device 100 having the characteristicsdescribed in Table 1 above. During the testing, (i) the pair of rotors114 (“rotor”) were oscillated in synchrony with each other, (ii) the armassembly 110 (“arm”) oscillated the rotors 114 about an orthogonal axis,and (iii) the oscillation of the arm assembly 110 was in synchrony withthe oscillation of the rotors 114. Further, the arm assembly 110 and therotors 114 were oscillated with the same amplitude and frequency and inphase with each other. The target position, load position, and currentwere recorded for the arm assembly 110, the rotors 114, and thecontrollers (e.g., the controller 148) for the first shaft 120(“spindle”). The target position was the position that the controller148 was programmed to attain, and the load position was the actualposition of the motor at the time of the sample.

The oscillations generated output gyroscopic torque about the axis A₁ ofthe first shaft 120 (e.g., the spindle). The rotating first shaft 120fed back gyroscopic and centrifugal torques to amplify the amplitude ofoscillation of the arm assembly 110, and the rate of rotation of thefirst shaft 120 was adjusted for centrifugal resonance. Under someparameters, the mechanical power output appeared greater than themechanical power input.

Referring to FIG. 14A, the arm motor controller reported the currentprovided by the controller every 4 milliseconds. The average current(Amp) is the current samples averaged over an integer number ofoscillation cycles in amperes. The fact that this current is negativemeans that arm is moving in the opposite direction as it normally movesfor a current of this sign. The rms current (A) is the root-mean-squarevalue of the recorded current. This number was used to calculate the PRpower (watts). The average power (watts) is the torque (N-m) multipliedby the speed (rad/s). The torque was calculated from the current bymultiplying the current by Kt which is the torque constant 0.34 N−m/Amp.In this graph, the average power is negative which means that themechanical power out of the arm is more than the mechanical power (e.g.,average current) that the motor controller puts into the arm. FR (watts)is the power dissipated in the windings of the motor. This is calculatedfrom the rms current squared times the winding resistance.

Referring to FIG. 14B, this graph shows data for one of the rotors atthe same settings as the previous graph of arm measurements. The datafor position and current were taken at 10 millisecond intervals by therotor motor controller. The rotor ave cur(A) line is the average of thecurrent measured by the rotor motor controller over an integral numberof cycles of the oscillation frequency. The rms cur(A) line is theroot-mean-square value of the rotor motor current. This was used tocalculate the I²R power that was dissipated in the resistance of themotor windings. The ave power(watts) line is the average of the torquetimes the velocity of the rotor averaged over an integral number ofcycles of the oscillation frequency. The torque was calculated from thecurrent by multiplying the current in amperes by the torque constant ofthe motor Kt which is 0.06 N/Amp for the rotor motors.

Referring to FIG. 14C, the spindle motor controller logged position andcurrent at 4 milliseconds per sample. The ave cur(A) line is the averageof the current over an integral number of cycles of the oscillationfrequency. The rms cur(A) line is the root-mean-square of the current.Since the spindle current was always in the same direction, the rms andthe average are very close to each other. The ave pwr (watts) line isthe torque times the speed of the spindle. The torque is calculated fromthe current by multiplying the current by Kt which is 0.34 N-m/amp. TheI²R pwr (watts) is the power dissipated in the resistance of thewindings of the motor.

Notably, the power required of the spindle motor decreased as theamplitude of the oscillation increased. The arm motor also showed adecrease. The rotor motors showed an increase that may, but notnecessarily, offset this decrease. If the decrease in spindle power withincreasing amplitude is further confirmed, then this can be seen asevidence that the oscillation of the arm and rotors results in powerbeing added to the spindle.

FIGS. 15A-15D are graphs illustrating sample test results for theeffects of oscillation of the rotors 114 on movement of the arm assembly120 of the centrifugal gyroscopic device 100. These effects were studiedunder two different conditions. First, the arm was held in a positionwith the motor controller while the rotor was oscillated. The currentdrawn by the arm motor was used to calculate the torque on the arm. Inthe second method, the arm motor controller was loaded with code whichhad the PID gains all set to zero. This allowed the position and currentdata to be recorded but the controller did not react to the position.The current in this case is the current generated by the motor when itmoved under the influence of the rotor motions.

Referring to FIG. 15A, this graph shows the arm being held in positionby the arm motor. During testing, the position varied only a smallamount (+−0.3 degrees) and the speed was nearly zero. This is the torquecalculated from the current by multiplying the current by Kt (0.34N−m/amp). The data shows a gyroscopic torque being applied to the armdue to the oscillation of the rotor combined with the rotation of thespindle.

Referring to FIG. 15B, in this case the arm was left free to move andthe arm motor controller was loaded with PID gains of zero so that themotion and current could be recorded. This graph shows the arm beingoscillated by the influence of the rotor oscillations and spindlerotation.

Referring to FIG. 15C, the torque applied to the arm was calculated fromthe current in the arm by multiplying by Kt (0.34 N−m/Amp) of the armmotor.

Referring to FIG. 15D, this graph shows a plot of torque versus speed toillustrate whether the device is consuming power or producing power. Ifthe signs of the torque and the speed are the same, the motor istransferring power to the device. If the signs are different, then thedevice is transferring power to the motor. In the case of thecontrollers, this transfer of power to the motor may have been wasted byproducing heat. In this case the speed and torque have opposite signs,implying that the arm motor is not transferring energy to the device.

FIGS. 16A-16F are graphs illustrating sample test results for theeffects of oscillation of the arm assembly 120 on the torque of therotors 114 of the centrifugal gyroscopic device 100. When the data wascollected for the effect of the rotor on the arm motions, the positionand current data for the rotor was also collected. This data wascollected to determine if the rotors were supplying energy to the arm.

Referring to FIG. 16A, the data in this graph shows one of the rotorsoscillating between 0 and 72 degrees. The actual position was arbitraryand was an artifact of the algorithm for generating the oscillation. Theoscillation of the rotor can be considered to be oscillating +/−36degrees. The actual oscillation amplitude is slightly larger than thisand the phase lags a little from the programmed target position. In thiscase, the arm was held in a nearly fixed position while the rotoroscillated, and the spindle rotated.

Referring to FIG. 16B, the rotor amplitude decreased when the arm wasfree to move compared to when the arm was held in position by the armmotor. This could be due to some energy being transferred from the rotorto the arm.

Referring to FIG. 16C, this graph shows current and speed data for therotor with the arm held in position.

Referring to FIG. 16D, this graph shows the current and speed in therotor with the arm free to move. The current in the arm was less whenthe arm is free to move. The graph of torque versus speed shows whetherthe rotor motor is transferring energy to the device or if the device istransferring energy to the motor. When the signs are the same, the motoris transferring energy to the device. When the signs are opposite, thetransfer is the other way.

Referring to FIG. 16E, in the case where the arm was held in position bythe arm motor, the resulting motion was an ellipse with an equal amountof energy transferred in both directions.

Referring to FIG. 16F, in the case where the arm was free to move, thisgraph is tilted to put more points in the two quadrants where the signsof the speed and torque are the same. This indicates that more energy isbeing transferred from the rotor motor to the rotor than is beingtransferred from the rotor to the motor. This is evidence that the rotormotor is providing energy to the arm motion. Notably, the amplitude ofthe current oscillation is not the factor that determines the amount ofpower transferred, but the relative phase of the torque and speedcurves, which is confirmed by the fact that the amplitude of the currentoscillation was greater in the case when there was little energytransfer from the motor and the amplitude of the current oscillation wassmaller in the case when the transfer of energy from the motor to therotor was greater.

IV. Select Embodiments of Additional Representative Characteristicsand/or Operational Parameters of Representative Elements of CentrifugalGyroscopic Devices

Table 3 below provides representative characteristics and/or operationalparameters of various elements of the centrifugal gyroscopic devicesdescribed herein.

TABLE 3 Representative Element of a Centrifugal Gyroscopic Embodimentsof the Present Device in Accordance with Representative Characteristicsand/or Operational Technology Parameters Rotors (e.g., PivotableMovement: +/− 20 degrees at 15 Hertz. Rotors 114) Inertia: 12,112.5grams × cm² (+/−1%). Geometry: balanced/symmetric around its axis ofrotation with majority of mass at outer ring. Position on Rotor Arm:rotors positioned at a distance at least 2× the radius of the rotor fromthe center of the rotor arm. Average Input Power: 97 watts. Peak InputPower: 195 watts. Rotor Mass: mass of the rotor minimized to increasethe rotor arm inertia, and forces on the rotor axis gearbox. E.g., 260grams. Rotor Diameter: diameter of the rotor impacts the length of therotor arm due to the requirement of distance from rotor arm center asset forth above. In an example configuration, the motors to generate therotor motion are straight-line coupled to the rotors. They are also kepttoward the center of the arm to reduce rotor arm inertia. Rotor Motor:Kollmorgen BLDC motor modeled in example as: AKM2G-33PL (48 VDC)/LowVoltage DC Drive with a 5:1 gearbox. Rotor Gearbox Axial Force: If thespindle is turning at 900 rpm, the rotor is at radius 25.66, traveling54 mph, and pulling outward with 133 lb-f or 593 Newton. In this exampleconfiguration the rotor is mounted directly to a gearbox, the bearingsof the gearbox would be subjected to that axial force. Rotor GearboxRadial Force: The rotor arm can accelerate/sweep through 21.5 degrees in0.0166 second. The rotor can be 25.7 centimeters from the center of therotor arm. The chord length is about 0.096 meter. Assuming a triangularacceleration profile, the acceleration of the rotor is 691.2 meters persecond squared, causing a 180 Newton force on the rotor. The distancefrom the gearbox flange to the rotor center plane can be 5 mm, such thatthe gearbox tilting torque is 180 N × 0.005 M = 0.9 N-Meter. Rotor DriveCentrifugal Force: An example of the forces generated by the motorgearbox combination: Total Mass: 2300 gram/5.07 lb (1400 grams motor,900 grams gearbox). Center of Mass: 13.46 cm from center of arm.Spinning at 900 RPM the motor/gearbox will pull on the coupling tubewith 618.2 lb-f/ 2750 Newton. Combined with the rotor pull force of (133lb-f or 593 Newton) results in 751.2 lb-f 3,343 Newton at each end ofcenter coupler. Rotor Arm Pivotable Movement: +/−43 degrees at 15 Hertz.(e.g., Arm Rotor Arm Motor: Assembly 110) Rotor Arm Assist Mode:AKM2G-44NL (96 VDC)/Low Voltage DC Drive with a 3.2:1 gearbox. Rotor ArmStart Mode: AKM2G-44NL (96 VDC)/Low Voltage DC Drive with a 12:1gearbox. Average Input Power: 524 watts. Peak Input Power: 1050 watts.Spindle Shaft Rotation: 15 revolutions per second or 900 rotations per(e.g., Spindle minute (RPM). 120) Mass: 33,504 grams/74 pounds. Inertiaabout “Z” axis: 17,128,266 grams × cm². Spindle Arm: AKM2G-31ML (96VDC)/Low Voltage DC Drive with an 8:1 gearbox. Average Input Power: 16watts. Peak Input Power: 125 watts. Power Source The device power willnot require 3-phase, not need more (e.g., Power than 240 VAC splitphase, can operate off standard 120 VAC Source 142) 10-15 Ampresidential AC outlet. Regeneration Predominantly the motion consideredis accelerating and To provide maximum efficiency, energy can becaptured during braking/decelerating in capacitors on DC Bus that willprovide energy for harvest on next acceleration move. RotorRegeneration: Rotor Peak Velocity: 400 RPM. Rotor Inertia: 12,112.5 gram× cm² ∥ 0.00121125 KgM². Braking Joules: 1.0626 J. deceleratingcyclically. Rotor Arm Regeneration: Rotor arm Peak Velocity: 428 RPM.Inertia handled by motor: 59,556 gram × cm² ∥ 0.00596 KgM². BrakingJoules: 5.986 J. Total Regeneration: For a total of ~14 Joules(Watt-Seconds) to be captured in ¼ cycle (1/60) so average of about 845watts for 0.0167 seconds.

V. Additional Examples

The following examples are illustrative of several embodiments of thepresent technology:

-   -   1. A centrifugal gyroscopic device, comprising:    -   a shaft rotatable about a first axis;    -   an arm coupled to the shaft and configured to rotate with the        shaft, wherein the arm is pivotable about a second axis        different from the first axis;    -   at least one rotor coupled to the arm and configured to pivot        with the arm about the second axis, wherein the at least one        rotor is further pivotable about a third axis different from the        first axis and different from the second axis; and    -   a control system operably coupled to at least one of the shaft,        the arm, and the at least one rotor, wherein the control system        is configured to bring the shaft, the arm, and the at least one        rotor into at least approximately a resonant mode of operation        in which (a) the shaft rotates about the first axis at a        rotational rate, (b) the arm oscillates about the second axis at        a first frequency, and (c) the at least one rotor oscillates        about the third axis at a second frequency at least        approximately equal to the first frequency.    -   2. The centrifugal gyroscopic device of example 1 wherein the        first frequency and the second frequency are at least        approximately equal to the rotational rate.    -   3. The device of example 1 or example 2 wherein the control        system includes a motor assembly positioned to drive the at        least one rotor to oscillate about the third axis at the second        frequency.    -   4. The device of any one of examples 1-3 wherein the oscillation        of the arm and the oscillation of the at least one rotor        generate a gyroscopic torque which acts to rotate the shaft        about the first axis, and wherein the gyroscopic torque is        substantially maximized in the resonant mode of operation.    -   5. The device of example 4 wherein the control system is        configured to change a phase relationship between the first        frequency of the arm and the second frequency of the at least        one rotor to change an average value of the gyroscopic torque.    -   6. The device of example 4 or example 5 wherein the control        system further includes a motor assembly positioned to drive the        arm to oscillate about the second axis at the first frequency.    -   7. The device of example 6 wherein the control system is        configured to control the motor assembly to change a phase        relationship between the first frequency of the arm and the        second frequency of the at least one rotor to change an average        value of the gyroscopic torque.    -   8. The device of any one of examples 1-7 wherein the rotation of        the arm generates a centrifugal force that acts to oscillate the        arm about the second axis.    -   9. The device of any one of examples 1-8, further comprising a        power generator coupled to the shaft and configured to generate        an output power from the rotation of the shaft.    -   10. The device of any one of examples 1-9, further comprising a        power generator coupled to the arm and configured to generate an        output power from the pivotable motion of the arm.    -   11. The device of any one of examples 1-10 wherein the second        axis is orthogonal to the first axis.    -   12. The device of any one of examples 1-11 wherein the third        axis is orthogonal to the second axis.    -   13. The device of any one of examples 1-12 wherein the at least        one rotor includes a first rotor coupled to a first end portion        of the arm and a second rotor coupled to a second end portion        the arm.    -   14. A method of operating a centrifugal gyroscopic device, the        method comprising:    -   rotating a shaft of the centrifugal gyroscopic device about a        first axis;    -   pivoting an arm of the centrifugal gyroscopic device about a        second axis different from the first axis, wherein the arm is        pivotably coupled to the shaft and configured to rotate with the        shaft;    -   pivoting at least one rotor of the centrifugal gyroscopic device        about a third axis different from the first axis and different        from the second axis, wherein the at least one rotor is        pivotably coupled to the arm and configured to pivot with the        arm about the second axis; and    -   controlling the rotation of the shaft, the pivoting of the arm,        and/or the pivoting of the at least one rotor to bring the        shaft, the arm, and the at least one rotor into a resonant mode        of operation in which (a) the shaft rotates at a rotational        rate, (b) the arm oscillates about the second axis at a first        frequency, and (c) the at least one rotor oscillates about the        third axis at a second frequency at least approximately equal to        the first frequency.    -   15. The method of example 14 wherein the first frequency and the        second frequency are at least approximately equal to the        rotational rate.    -   16. The method of example 14 or example 15 wherein the method        further comprises generating a gyroscopic torque by the        oscillation of the arm and the oscillation of the at least one        rotor which acts to rotate the shaft about the first axis.    -   17. The method of example 16 wherein the method further        comprises changing a phase relationship between the first        frequency of the arm and the second frequency of the at least        one rotor to change an average value of the gyroscopic torque.    -   18. The method of any one of examples 14-17 wherein the method        further comprises generating power, with a power generator        coupled to the shaft, by the rotating the shaft.    -   19. The method of any one of examples 14-18 wherein the second        axis is orthogonal to the first axis, and wherein the third axis        is orthogonal to the second axis.    -   20. A centrifugal gyroscopic device, comprising:    -   a spindle rotatable about a first axis;    -   an arm coupled to the spindle and configured to rotate with the        spindle, wherein the arm has a first end portion and a second        end portion, and wherein the arm is pivotable about a second        axis different from the first axis;    -   a first rotor coupled to the first end portion of the arm;    -   a second rotor coupled to the second end portion of the arm,        wherein the first rotor and the second rotor are each pivotable        about a third axis different from the first axis and different        from the second axis; and    -   a control system operably coupled to at least one of the shaft,        the arm, the first rotor, and the second rotor, wherein the        control system is configured to bring the shaft, the arm, the        first rotor, and the second rotor into a resonant mode of        operation in which (a) the shaft rotates at a rotational        rate, (b) the arm oscillates about the second axis at a first        frequency at least approximately equal to the rotational rate,        and (c) the first rotor and the second rotor oscillate about the        third axis at a second frequency at least approximately equal to        the first frequency.

VI. Conclusion

The above detailed description of embodiments of the present technologyare not intended to be exhaustive or to limit the technology to theprecise forms disclosed above. Although specific embodiments of, andexamples for, the technology are described above for illustrativepurposes, various equivalent modifications are possible within the scopeof the technology as those skilled in the relevant art will recognize.For example, although steps are presented in a given order, otherembodiments may perform steps in a different order. The variousembodiments described herein may also be combined to provide furtherembodiments.

From the foregoing, it will be appreciated that specific embodiments ofthe technology have been described herein for purposes of illustration,but well-known structures and functions have not been shown or describedin detail to avoid unnecessarily obscuring the description of theembodiments of the technology. Where the context permits, singular orplural terms may also include the plural or singular term, respectively.

As used herein, the terms “about,” “approximately,” “generally”,“substantially,” and the like refer to values within 10% of the statedvalue. As used herein, the phrase “and/or” as in “A and/or B” refers toA alone, B alone, and A and B. To the extent any materials incorporatedherein by reference conflict with the present disclosure, the presentdisclosure controls. Additionally, the term “comprising” is usedthroughout to mean including at least the recited feature(s) such thatany greater number of the same feature and/or additional types of otherfeatures are not precluded. It will also be appreciated that specificembodiments have been described herein for purposes of illustration, butthat various modifications may be made without deviating from thetechnology. Further, while advantages associated with some embodimentsof the technology have been described in the context of thoseembodiments, other embodiments may also exhibit such advantages, and notall embodiments need necessarily exhibit such advantages to fall withinthe scope of the technology. Accordingly, the disclosure and associatedtechnology can encompass other embodiments not expressly shown ordescribed herein.

I/We claim:
 1. A centrifugal gyroscopic device, comprising: a shaftrotatable about a first axis; an arm coupled to the shaft and configuredto rotate with the shaft, wherein the arm is pivotable about a secondaxis different from the first axis; at least one rotor coupled to thearm and configured to pivot with the arm about the second axis, whereinthe at least one rotor is further pivotable about a third axis differentfrom the first axis and different from the second axis; and a controlsystem operably coupled to at least one of the shaft, the arm, and theat least one rotor, wherein the control system is configured to bringthe shaft, the arm, and the at least one rotor into at leastapproximately a resonant mode of operation in which (a) the shaftrotates about the first axis at a rotational rate, (b) the armoscillates about the second axis at a first frequency, and (c) the atleast one rotor oscillates about the third axis at a second frequency atleast approximately equal to the first frequency.
 2. The centrifugalgyroscopic device of claim 1 wherein the first frequency and the secondfrequency are at least approximately equal to the rotational rate. 3.The device of claim 1 wherein the control system includes a motorassembly positioned to drive the at least one rotor to oscillate aboutthe third axis at the second frequency.
 4. The device of claim 1 whereinthe oscillation of the arm and the oscillation of the at least one rotorgenerate a gyroscopic torque which acts to rotate the shaft about thefirst axis, and wherein the gyroscopic torque is substantially maximizedin the resonant mode of operation.
 5. The device of claim 4 wherein thecontrol system is configured to change a phase relationship between thefirst frequency of the arm and the second frequency of the at least onerotor to change an average value of the gyroscopic torque.
 6. The deviceof claim 4 wherein the control system further includes a motor assemblypositioned to drive the arm to oscillate about the second axis at thefirst frequency.
 7. The device of claim 6 wherein the control system isconfigured to control the motor assembly to change a phase relationshipbetween the first frequency of the arm and the second frequency of theat least one rotor to change an average value of the gyroscopic torque.8. The device of claim 1 wherein the rotation of the arm generates acentrifugal force that acts to oscillate the arm about the second axis.9. The device of claim 1, further comprising a power generator coupledto the shaft and configured to generate an output power from therotation of the shaft.
 10. The device of claim 1, further comprising apower generator coupled to the arm and configured to generate an outputpower from the pivotable motion of the arm.
 11. The device of claim 1wherein the second axis is orthogonal to the first axis.
 12. The deviceof claim 1 wherein the third axis is orthogonal to the second axis. 13.The device of claim 1 wherein the at least one rotor includes a firstrotor coupled to a first end portion of the arm and a second rotorcoupled to a second end portion the arm.
 14. A method of operating acentrifugal gyroscopic device, the method comprising: rotating a shaftof the centrifugal gyroscopic device about a first axis; pivoting an armof the centrifugal gyroscopic device about a second axis different fromthe first axis, wherein the arm is pivotably coupled to the shaft andconfigured to rotate with the shaft; pivoting at least one rotor of thecentrifugal gyroscopic device about a third axis different from thefirst axis and different from the second axis, wherein the at least onerotor is pivotably coupled to the arm and configured to pivot with thearm about the second axis; and controlling the rotation of the shaft,the pivoting of the arm, and/or the pivoting of the at least one rotorto bring the shaft, the arm, and the at least one rotor into a resonantmode of operation in which (a) the shaft rotates at a rotational rate,(b) the arm oscillates about the second axis at a first frequency, and(c) the at least one rotor oscillates about the third axis at a secondfrequency at least approximately equal to the first frequency.
 15. Themethod of claim 14 wherein the first frequency and the second frequencyare at least approximately equal to the rotational rate.
 16. The methodof claim 14 wherein the method further comprises generating a gyroscopictorque by the oscillation of the arm and the oscillation of the at leastone rotor which acts to rotate the shaft about the first axis.
 17. Themethod of claim 16 wherein the method further comprises changing a phaserelationship between the first frequency of the arm and the secondfrequency of the at least one rotor to change an average value of thegyroscopic torque.
 18. The method of claim 14 wherein the method furthercomprises generating power, with a power generator coupled to the shaft,by the rotating the shaft.
 19. The method of claim 14 wherein the secondaxis is orthogonal to the first axis, and wherein the third axis isorthogonal to the second axis.
 20. A centrifugal gyroscopic device,comprising: a spindle rotatable about a first axis; an arm coupled tothe spindle and configured to rotate with the spindle, wherein the armhas a first end portion and a second end portion, and wherein the arm ispivotable about a second axis different from the first axis; a firstrotor coupled to the first end portion of the arm; a second rotorcoupled to the second end portion of the arm, wherein the first rotorand the second rotor are each pivotable about a third axis differentfrom the first axis and different from the second axis; and a controlsystem operably coupled to at least one of the shaft, the arm, the firstrotor, and the second rotor, wherein the control system is configured tobring the shaft, the arm, the first rotor, and the second rotor into aresonant mode of operation in which (a) the shaft rotates at arotational rate, (b) the arm oscillates about the second axis at a firstfrequency at least approximately equal to the rotational rate, and (c)the first rotor and the second rotor oscillate about the third axis at asecond frequency at least approximately equal to the first frequency.